Calibrated irreducible representations and Jacobson radicals of the periplectic Brauer algebra $A_n$ and its degenerate affine extension $\mathrm{s}\mathbb{W}_n$

The periplectic Brauer algebra $A_n$ , the diagram algebra governing centralizer algebras of the periplectic Lie superalgebra $\mathfrak{p}(n)$ , is intrinsically non-semisimple, so the classical Okounkov–Vershik approach to the symmetric group — which rests on modules diagonalizable under Jucys–Murphy elements — does not automatically apply. Im and Norton constructed an explicit tower of irreducible calibrated (Jucys–Murphy-diagonalizable) $A_n$ -modules $\mathrm{C}_0(\Lambda^k V_n)$ on which, for $1\le k\le n-2$ , the cup-cap generators act nontrivially, computed their full Jucys–Murphy spectrum, identified the radicals in small cases, and posed three open problems: classify all irreducible calibrated $A_n$ -modules (in particular, decide whether any exist beyond the symmetric-group inflations and the constructed exterior-power family); rigorously identify the constructed modules within the cell-module labeling $\mathrm{L}(\lambda)$ and reconcile the conflicting eigenvalue branching rules; and determine the Jacobson radicals $J(A_n)$ and $J(\mathrm{s}\mathbb{W}_n)$ for general $n$ , including whether Coulembier's central element $\Theta$ is nonzero. All three questions remain open.

Background

The periplectic Lie superalgebra $\mathfrak{p}(n)$ is the Lie superalgebra of endomorphisms of $\mathbb{C}^{n|n}$ preserving an odd non-degenerate symmetric bilinear form. Its invariant theory is governed, in analogy with how the Brauer algebra (the classical diagram algebra, spanned by perfect matchings, that describes the centralizer algebras of orthogonal and symplectic groups) governs those groups, by the periplectic Brauer algebra $A_n$ : a finite-dimensional algebra over $\mathbb{C}$ spanned by Brauer diagrams — perfect matchings on $2n$ points arranged in two rows of $n$ , composed with sign rules that make the resulting diagram category a monoidal supercategory. It is generated by elements $s_1,\dots,s_{n-1}$ obeying the Coxeter relations of the symmetric group $S_n$ together with nilpotent cup-cap generators $e_1,\dots,e_{n-1}$ satisfying $e_i^2=0$ and $e_{i\pm1}e_ie_{i\pm1}=-e_{i\pm1}$ , plus mixed relations among the $s_i$ and $e_i$ — in the conventions of Im and Norton these read $e_is_i=e_i$ , $s_ie_i=-e_i$ , and $s_ie_{i+1}e_i=s_{i+1}e_i$ , with signs opposite to the conventions of Coulembier and of Chen–Peng, a discrepancy relevant to component (2) of the problem below. Unlike the Brauer algebra at generic parameter, $A_n$ is never semisimple (Coulembier, Proc. Lond. Math. Soc. 2018). Coulembier equipped $A_n$ with a cellular structure: cell modules $\mathrm{W}(\lambda)$ are labeled by partitions $\lambda$ of $n, n-2, n-4,\dots$ , and the simple modules are the quotients $\mathrm{L}(\lambda)=\mathrm{W}(\lambda)/\mathrm{rad}\,\mathrm{W}(\lambda)$ ; at even $n$ , $\varnothing$ labels no simple module in this classification (whether the cell module $\mathrm{W}(\varnothing)$ at those levels is nonetheless irreducible is part of question (2) of the Problem Statement below). Coulembier also introduced the periplectic Jucys–Murphy elements, defined recursively by $Y_1:=0$ and $Y_{j+1}:=s_jY_js_j+s_j+e_j$ , which generalize the classical Jucys–Murphy elements of $\mathbb{C}S_n$ (the transposition sums $\sum_{j<i}(j\,i)$ , on whose simultaneous diagonalizability the Okounkov–Vershik approach to $\mathbb{C}S_n$ -representation theory rests) and generate a commutative subalgebra of $A_n$ , and he classified the blocks in characteristic zero; Coulembier and Ehrig computed the composition multiplicities of the cell modules, and Barbier–Cox–De Visscher later determined the blocks in positive characteristic. A representation of $A_n$ is called calibrated if it admits a basis of simultaneous eigenvectors for all of $Y_1,\dots,Y_n$ — a Gelfand–Tsetlin-type condition that every irreducible $\mathbb{C}S_n$ -module satisfies by the Okounkov–Vershik theory, but which is exceptional for the non-semisimple $A_n$ . Two calibrated families are known: the inflations of irreducible $\mathbb{C}S_n$ -modules (on which every $e_i$ acts by $0$ ), and the modules constructed by Im and Norton. To build the latter they used the degenerate affine periplectic Brauer algebra $\mathrm{s}\mathbb{W}_n$ (defined by Chen–Peng and by Balagović, Daugherty, Entova-Aizenbud, Halacheva, Hennig, Im, Letzter, Norton, Serganova and Stroppel), which extends $A_n$ by commuting polynomial generators $y_1,\dots,y_n$ interacting with $s_i,e_i$ through the same relations the $Y_j$ satisfy — the periplectic analogue of the degenerate affine Hecke algebra in $S_n$ -theory. Im and Norton proved that on $\Lambda^k V_n$ , the $k$ -th exterior power of the $(n-1)$ -dimensional standard representation $V_n$ of $S_n$ , explicit formulas for the $s_i$ , $e_i$ and $y_j$ (the parameter $\alpha$ being the eigenvalue of $y_1$ ) define an irreducible calibrated $\mathrm{s}\mathbb{W}_n$ -module $\mathrm{C}_\alpha(\Lambda^k V_n)$ , which for $\alpha=0$ factors through $A_n$ and yields irreducible calibrated modules $\mathrm{C}_0(\Lambda^k V_n)$ on which every $e_i$ acts by a nonzero matrix for $1\le k\le n-2$ , while the endpoint modules $\mathrm{C}_0(\Lambda^0V_n)$ and $\mathrm{C}_0(\Lambda^{n-1}V_n)$ are the trivial and the sign representations of $S_n$ , with the $e_i$ acting by $0$ ; the eigenvalues of $Y_{j+1}$ differ from those of $Y_j$ by $\pm 1$ , and as $S_n$ -modules these are exactly the hook representations (those labeled by hook partitions $(n-k,1^k)$ , a single row together with a single column). Under restriction along the tower $A_1\subset A_2\subset\cdots$ , these modules form a non-semisimple categorification of Pascal's triangle. On the radical side: the Jacobson radical $J(A)$ of an algebra $A$ is the intersection of all its maximal left ideals — for a finite-dimensional algebra such as $A_n$ equivalently its largest nilpotent two-sided ideal, and the obstruction to semisimplicity; known cases are $J(A_2)=\mathbb{C}e_1$ and the $5$ -dimensional $J(A_3)$ computed by Moon in early work on the algebra, while for $n>3$ Coulembier's central element $\Theta=\prod_{2\le i<j\le n}\bigl(1-(Y_i-Y_j)^2\bigr)$ and all its multiples $\Theta f$ by symmetric polynomials in $Y_2,\dots,Y_n$ lie in $J(A_n)$ but might be zero, and Im and Norton determined $J(\mathrm{s}\mathbb{W}_2)=(e_1)$ with upper bound $J(\mathrm{s}\mathbb{W}_n)\subsetneq(e_1)$ for $n>2$ and lower-bound elements $e_1e_3\cdots e_{n-1}\in J(A_n)\cap J(\mathrm{s}\mathbb{W}_n)$ for even $n$ .

Problem Statement

Develop the Okounkov–Vershik-type representation theory of the periplectic Brauer tower $A_1\subset A_2\subset\cdots$ by solving the three linked problems posed by Im–Norton (2020). (1) Classify the irreducible calibrated representations of $A_n$ for every $n$ : determine whether every irreducible $A_n$ -module admitting a basis of simultaneous eigenvectors for the Jucys–Murphy elements $Y_1,\dots,Y_n$ is, up to isomorphism, either an inflation of an irreducible $\mathbb{C}S_n$ -module (on which all $e_i$ act by $0$ ) or one of the constructed modules $\mathrm{C}_0(\Lambda^k V_n)$ ; equivalently, decide whether there exists an irreducible calibrated module on which some $e_i$ acts nonzero but for which some joint eigenvalue tuple $(\alpha_1,\dots,\alpha_n)\in\mathrm{Spec}(L)$ , the tuple of eigenvalues of $Y_1,\dots,Y_n$ on a simultaneous eigenvector, has a consecutive difference $\alpha_j-\alpha_{j+1}\notin\{+1,-1\}$ , and exhibit or exclude all such modules. (2) Identify the constructed tower inside the cell-module labeling. This component consists of three independently-standing sub-claims: (2a) prove or disprove that, in the labeling conventions of the cellular structure and possibly up to transpose, $\mathrm{C}_0(\Lambda^k V_n)\cong\mathrm{L}(n-2k)$ (a one-row partition) for $k\le\tfrac{n-1}{2}$ and $\mathrm{C}_0(\Lambda^k V_n)\cong\mathrm{L}(1^{\,2(k+1)-n})$ (a one-column partition) for $k\ge\tfrac{n-1}{2}$ ; (2b) exhibit an explicit identity or map relating the cell-module eigenvalue branching rule for up-down tableaux in the Bratteli diagram (the graph recording restriction of irreducibles along the tower) — eigenvalue $\mathrm{ct}(b)+1$ upon removing a box $b$ of content $\mathrm{ct}(b)$ , its column index minus its row index — to the sign rule $\mathrm{ct}(b)\pm1$ observed for the constructed tower; and (2c) settle whether the cell module $\mathrm{W}(\varnothing)$ at even $n$ is irreducible and isomorphic to $\Lambda^{n/2}V_n$ . (3) Determine the Jacobson radicals $J(A_n)$ and $J(\mathrm{s}\mathbb{W}_n)$ for general $n$ : decide whether Coulembier's central element $\Theta=\prod_{2\le i<j\le n}\bigl(1-(Y_i-Y_j)^2\bigr)$ is nonzero in $A_n$ for $n>3$ , describe the ideals $\Theta\,\mathbb{C}[Y_2,\dots,Y_n]^{S_{n-1}}$ and the radical precisely in terms of the diagram generators (improving the known inclusions and bounds), and thereby compute the maximal semisimple quotients $A_n/J(A_n)$ and the radical of the degenerate affine algebra $\mathrm{s}\mathbb{W}_n$ . These three components jointly support one objective — a complete calibrated/Gelfand–Tsetlin layer plus radical structure for the non-semisimple tower — with the identification question (2) serving as the bridge between the constructed calibrated family and the cell-module labeling required by the classification (1).

References

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